On a diphasic low mach number system

¹ Commissariat à l'Énergie Atomique, 91191 Gif sur Yvette, France.
² Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128-succursale Centre-ville, Montréal, H3C 3J7, Canada. stephane.dellacherie@cea.fr; dellache@crm.umontreal.ca

Abstract

We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech. 42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van der Waals equations of state, we recover some natural properties related to the dilation and to the compression of bubbles. We also propose an entropic numerical scheme in Lagrangian coordinates when the geometry is monodimensional and when the two fluids are perfect gases. At last, we numerically show that the DLMN system may become ill-posed when the entropy of one of the two fluids is not a convex function.